Deposition of steeply infalling debris around white dwarf stars

High-metallicity pollution is common in white dwarf (WD) stars hosting remnant planetary systems. However, they rarely have detectable debris accretion discs, possibly because much of the influx is fast steeply infalling debris in star-grazing orbits, producing a more tenuous signature than a slowly accreting disc. Processes governing such deposition between the Roche radius and photosphere have so far received little attention and we model them here analytically by extending recent work on sun-grazing comets to WD systems. We find that the evolution of cm-to-km size (a(0)) infallers most strongly depends on two combinations of parameters, which effectively measure sublimation rate and binding strength. We then provide an algorithm to determine the fate of infallers for any WD, and apply the algorithm to four limiting combinations of hot versus cool (young/old) WDs with snowy (weak, volatile) versus rocky (strong, refractory) infallers. We find: (i) Total sublimation above the photosphere befalls all small infallers across the entireWD temperature (TWD) range, the threshold size rising with TWD and 100x larger for rock than snow. (ii) All very large objects fragment tidally regardless of TWD: for rock, a(0) >= 10(5) cm; for snow, a(0) >= 10(3)-3 x 10(4) cm across all WD cooling ages. (iii) A considerable range of a(0) avoids fragmentation and total sublimation, yielding impacts or grazes with cold WDs. This range rapidly narrows with increasing TWD, especially for snowy bodies. Finally, we briefly discuss how the various forms of deposited debris may finally reach the photosphere surface itself.